lollipop sticks provide plenty of support for cake pops (even larger then average pops). and lollipop sticks come in various lengths.

I like to use an assortment of sizes of lollipop sticks for a display, sizes 8, 6, and 4 inch heights.

I've never tried mounting a cake pop on a cookie stick, but I would image the larger width might cause the cake pop to split.

the only caution I can think of would be using a really long lollipop stick...if the stick is waved in the air (like in the hands of an excitable child) then the cake pop might be launched off the stick and go airborn (WEEEeeeee )